I. Modifications of the Radial Distribution Method of Interpretation of Electron Diffraction Photographs of Gas Molecules. II. The Electron Diffraction Investigation of the Structure of Benzene, Pyridine, Pyrazine, Butadine, Cyclopentadiene, Furan, Pyrro

I. f;~ODIFICL':HOns OF RADil:.L DISTRIB1J'I1 ION

OF

I!J'L'ERFjHETATIOH Of 15LECTRO:N Dili'FRAGTION OF

'

FU.RJ1.N, PYRROLE, AUD THIOPHEN.E

Thesis by

Verner Schomaker

In Partial Fulfillment of the Requirements for the Degree ot Doctor ot Philosophy

California Institute of Tecl:lnology Pasadena, Cal1fo rnia

I. i.:odifioaiiions of' the Radial Distribution 11Iethod

ot

Interpretation oi' ..i:leotron Di:t'f'raotio:c1 ?h.oto-gr'1-ph.a of Gas I:.;oleoules

11... lntrod·..:tc t ion B.. General :Z·heory

l~ .De:t'ini iiion o:t' D(.L') and D' ~);

:Je:r:Lva-tion of a general radial distribu:Je:r:Lva-tion

Page

1

formula 2

2. Substitution of a summation over

dis-orete ·terms for integr1.:1tion over all

values of" the angle variable ? 3. The "art ifioial temperature 1'aotor";

its relation to an only finite upper

limiting 'Talue of' the angle variable 14

C. A:p:pl ioa t ions

l. The radial distribution method

ot

Pauling and Brockway. The transition

to a modified method 28

2. The modified radial distribution method 51

3. Some special points 33

a4 The etreot of omitting the .first

terms of the summation; The

faotor l/~~,SJ 34

6..R"

b. The faotor _L'.ze- 36

a. Attenipted quan·t1tat1ve estimation

ot errors

arising

from (e) and (b);

comparison ·with errors of'

inte:r-atomio distances obtained by the

older radial distribution method 58

44 Discussion and suggested further

II. The Electron Diffraction Investigation of the

Jtruotilre of Benzene, Iyridene, Pyrszine,

DutaCliene 1 Cycilo:pentadiene .; .Uur:an, 3:yr:ro1e :1

Page

a.;_d 1.1hiophe:n.e 4 7

I l I. ::Jur.1.nrn:ry 59

I'l, ,h.clcnowledgment 60

V

Propositions 61

OF

OF

D

A. Introduot1on

The problem of the interpretation of data obtained

experim.ents on d ion ot eleotrons by

gas moleeules1, tor tlle :purpose of obtaining

tion about inter~tomia distanoea and _in

a large nUtnber of compounds, must be at·taolced in a

that is both etticient and reasona.bly acourate. exist. The first 1 oalled the '"visual'•

method2 , consists of the qualitative and quantitative comparison

ot

the diffraction pattern, as determined by visual inspection

ot

the photographic record~ with that calculated w1 th tne aid of a theo:r:etioal intensi t;r tunatlon tor an assumed molecular model. The second,

tlle "radial distribution"

method~\

employs the samei data as the first, but by an inversion of the theor• •tioal formula leads directly to the de$1red informa-tion. [ts advantages or sim.pl1o1ty and economy of

effort have, however, been

generally overbalanced by

the greater power and reliability ot the Visual method.

1.. For a. oomp:rehens ive :review on the investigation of structure of gas molecules by the eleotron dif•

traction method, see L.O. Brockway,

Rev.

Mod. Phys.t

§.,

231, ( l\136).

2. R. W1erl, Ann. d. Physik,

§.,

231, (1936).

L. Pauling and L.O. Brook.way, J. Ohem. Phys.,!'

86 7' ( l 934).

For relatively unimportant and little used

variants

ot

the method see Brockway, loo. oit.

2.

In this section modifications, involving'

v

little

additional labor, are presented, wnioh, to& \er with

the generally .more complete date avail~fble at ~esent,

greatly increase the e oaoy the :radial d

•1bu-tion :method, and render 1 t equal to the visual n... 'lod

in 1.m.portanoe as a tool for the treatment o:t' eleo\ n dittraotion data in the study

ot

molecular struotur

With these modifications, it can generally be de~endt

upon to give interatomic distances which suggest a model that needs only to be refined somewhat by the

application ot tne visu~l met.hod.

The greater part ot this section is devoted to

a discussion of the theory, developed Chiefly by

Dr. Degard, which is tound to underlie theee

modifica-tions, and 01' an effort to estimate the importanoe of

the approximations involved in the radial distribution

method as it now stands. The eommon errors of the older radial distribution method can be partially understood with the help of the conclusions based on

this discussion, and further modifications of the radial distrj.bution method are suggested.

3.

of this section, and to Dr. Simon Bauer for helpful

early d1soussione, and I owe thanks to Professor ?tn.J.ling, es:pec ially, for o onstant, and frequently nacessa1·y,

encouragement, :us well as

tor

valuable help '.11·~h the

B. General Theory

to

l. Definition of D(,,t'); Deriv~tion of the general radial distribution formula.

the purposes of 'this section 1 t is desirable

ze the usual expression for the l

t cone eleotrons elastically soat·

tared at an angle ,,..9 with resp:iot to an homogeneous incident beam of electrons of wave length .A by

randomly oriented gas molecules. 1

rh1s is done by

introducing a formally continuous distribution tunotion

D ~r)

cL)'

whioh is proportional to the probability of'

finding in the molecule and atom 1 and an atom j

e.

distance

~

e.part. The result is:

\ " " [ " " ' " l: I (Zi-fi/(Z:j-F7/ ~;;;-_, ( i

l

I

rs; = k' ~ .::i!iE'..-17 &l·.,.~.,/f'' z ... ;z.

.s-.,

.s-e

;'no/ecq/q,. , . .o 'i:7

.t~f·

.

,

where

S7T<G~£::t)~

k' is the uninteresting constant

ZL {

=Ila

,

z,

1s the atom10 number of the 1-th atom and F, , its

""" .,,. ,.a....;...9..4J ,tJ

X-ray form factor, and s

=

X

•

D<j (...<} 1• nol".U'.l.al1aed in the following manner.

( 2)

L

~"

p,;t.i'

Ju -

[D.:;

U.)

.l'.e

%

z.

Zj·

l. See,

tor

example, Brockway, Rev. Mod. Phys., ~'

It is olear tnat {l) automatically provides for the treatnent ot rigid, as well as non-rigid, molecules.

Equation {l) may ·be l"f!lwritten as

(3)

I

~.1==A''f;.f1~~::~~

f

LJ..;.r~

Q & > d . .__--, _ ___.,.

o f /

this is not enough tor our purpose; it is necessary to write the integrand of I(s) as a produot of the form,

utJ/

sin{s.f'L

?l(s).

It

1s

convenient

to

1ntroduoe

the

following definitions:

Evidently the desired separation oannot in general be

aooomplished, even if the structure of the m¢lecule

under consideration is known; and obviously, of course, it is impossible 1t nothing is known about the structure

ot the molecule at all. However, it is seen on

examina-tion of the form tao tors, F ,· , that

z~~;-·

is nearly l

the same function

ot

s tor all atoms , so that for any

set ot atol!ls 1t is possible ~to choose a function /{s)

which will be a reasonably good representation of /( s , / ) .

Usually, indeed it will be round satisfactory to ig,nore the variability of

-¥7 (

s).. As the result of all this,

(3) may now be written

{ 6

l

I

ts.J

=-

k

j

oa

~

-d'

LJI{//

~

..s-/ /...f" ()

is justitiablet as will be seen later, to

allow s to have all positive values instead of imposing

the functional limit which arises from its definition. This being true, (6) satisfies the requirements of a Fourier integral anct can be inverted in the usual

way.

1f1

9ru1=

J':-

f-/M~

a;t-

~,,.

satisfies certain requirements, and u, as well as

"t,

may have all positive values, then,

The application of these formulae to ( 6) yields the desired result2,

2 , where K-= 7ri<' •

1. Ooure.nt and Hilbert, ":Methoden der Mathematisohen Phys1k," Vol. I, Julius Springer, Berlin, page 68.

a.

This is a slight extension

ot

Degard's formula 3 (6.3)

D(

'_Pl=Kj~"JrsJ ~..:s~ d.s ·

1-t'atJt/l)/,f=J

/

0 £;z:-F,)z. .s--::R .I c

which is ap~licable to homoatomio molecules only, and is prob~bly. more generally usetul than the kind of treatment of (6.3) given by Degard in the oase of carbon tetrachloride.

7.

a.

Substitution of a summation over discrete term.a for the integration over all values of s.

In the radial distribution method of Pauling $nd

Brockway1 equation (7) is replaced by a sum, of whioh

eaoh term represents the part of the integral ove1· a

funotion. Although this prooedure was found to be

satisfactory, its use in calculating radial distribution

functions with visually estimated intensities did not

make clear the significance of the approximation

involved. This, the later wo1·lc of Degard has done .. Pegard carefully determined the relative molecular intensities as a function of s, and evaluated the

2

integral of the radial distribution funotion numerically • He tben showed that quite precisely the same result is given by the following sum, the terms of which are the same as those of the older formula except tor the

exponential faotor.

-ts)

DPJ=t;e~

i. J.1 ...

c.s.,

.§1., 2684, (1935).

I

Here the k-th term olosely approximates the part of the integral between the sub-11m1te s~ and s;

'

the zeros of' the molecular intensity function ad jaoent

to the

Vlidth

ot

the teature 1n question.

In

order to disoues the theoretical s1gn1f1ca.noe

ot

this result, to dis-cover its generalization for more complioated moleoules, Degard derived the mathematical expression for the

intensity function

wnioh

exactly corresponds to the ~adi.al distribution function ( 8).. In the next para-graph Dr. Degard's treatment is reproduoed··w1th some

simplification of the mathematical d1soussion, and otb.er slight changes.

The theoretical intensity tunotion which is related to (8) is found by substituting for D'(~) in (?) its expression by equation (8):

,,;-/r;,-s/j~

. / /

~

s,.e'..f

r

e.'4

e

-...e<./~

s.c_d' ?=.

'Z

IK'ts.J

( g)

I

f.s:,1 = _,..:r l J"' :a-,..P -5 - lo(

0

where

"'°-~.fit,

9.

By the well known integral,

(

""

0o _rzif'"

m-

- J A4

l

e

~_B'xd2¢

= ;e

*

e j fa.>o

1

1 (SJ

,

...

, this becomes

is negligible tor the values of '" and s whioh are

ot

interest. Therefore,

On multiplying (13) by

that the distribution of' intensity corresponding to (8)

is the same as the theoretical intensity function I(s) insofar as the reduced theoretical intensity function

(15 )

I

o

=~

fcs.1

=[d./L'oU/..~.s'_f

C .:S,) k I' 'I'S.) 0

oan be represented as a sum of Gaussian distributions:

(16)

It will be shown that this oan be done quite well by letting the

sK

be the a values of the maxima and min1.In.a at the reduced theoretical curve, and choosing

the b" and G ~ to correspond to their widths and

Degard used a term for each feature (maximum or

minimum) of the reduced theoretioal curve, and,

more-over, assumed that a :peak, eay, is 'pest represented

,,,

'

'"'K ional to the beigll't of the peak,

a11d bk.

=

CJ. 36/ (S"''-S,.,.)~ , that is to Se!.:;{, if the

Gaussian and the peak have the same halt-width {s,! ... sK) ..

Figure Ia shows this representation of the specially FvH.::1',.~h

simple reduced theoretioalA(sin s..B) for a rigid diatomic

molecule, where the o~ have been given the value

l.os.

It is seen that the theoretical ourve (dashed) and the sum ot the Gaussian representations {heavy, solid curve)

are indeed very nearly alike. Figure Ic shows the

representation whioh is obtained by using Gaussian

terms for the ridges only, in the following manner.

It is required that when the sum of the Gaussians ooinoides vvith the maxim.a and minima of the reduoed theoretieal ourve (after subtracting a constant,

non-essential part), tbe representation shall as

closely as possible approach the theoret1oal curve at

its points of inflection, which, if we now regard it

as a series of peaks, are the points of halt-maxi~al

c.s-&"")

ordinates. If we write:

I"

cs.J =

~

e,.,.' e

-4

and their best solution gives

fl1e' = 0 _;

'Xhe representation fails in that 1 t has a too sraall

ordinate at points corresponding to the inflections

ot the theoretical ourve. Gl:\.o1oes of bt< from 0Asr.s-,/-s1eJJ!.

to o.fiocs:-s,.,f· _{ere very nearly (within 2%} as good}

as this optimum. For representations with both maxima

and i t seems 1·eason.able to demand that when the

sum of the Gaussian terms fits the reduced theoretical

curve at its maxim&, minima, and nodes, it should in

addition fit as closely as possible at the half-maximal

points of the ridges and valleys. The problem of find-ing the best value of bK turns out to be exactly the

same as before, and the result is formally the sarne;

one rn.ust only place the required i11te:r11Tetatio:r1 on the term "half-width." The representation obtained with

this clloioe o:t' bl!( is illustrated in Figure Ib, where :Lt should be noticed that a slight ohange ot scale

would aex;ve to show that this ohoioe of b" does give

l. In formulating these conditions it haa been assumed

that the Gaussian has a negligible value when its argument is equal to tour times its halt-width.

Al-thousn this assumption :is not preoisely t:rue, 1 t has

no appreciable effect on the determination of b~. The assumption nas not been made in plotting the

a better representation than that at Degard, although

the difference is probably not s _fioant.

The p:raotioal problem represen~ing the reduced

d:l to Clisouss. r·t _{to remark}

·that th.e ve1·~r good representation obtainatlle for the

simple sine function indicates that the SUl'.ll of Gaussian

ter:rna is probably suitable for representing the more

oo:mp11os:ted Fourier series also. Probably it is best

always to represent both :maxima and minima of the

Fourier series in the hope of obtaining :f'rom tne sum

of two fairly suitable representations one that is

somewhat better than either alone. Moreover, it seems

that relatively small values

ot

b*< should be taken,

when the representation of both maxima and minima

used, in order to reduce the overlapping of adjacent

Gaussian terms, and consequently also the dependence

on their nioe oanoellation. For in the more oomplicated

oases it oannot be expected that this oancellat1on will

be as satisfactory as 1. t is in the example discussed

above. Asymmetric peaks, or peaks with "shelves" are

apeeittlly difficult. Generally ~ term with a emall

peak, and to the shelf can be taken to represent a

composite feature of the intensity function,.

In this section it has beeri proved ·that the

radial distribution function can be replaced

a S't..llll

(8) _{b is}

equivalent

to it insofar as the reduced

intensity

tunotion oan be represented by a su.m of Gaussian ternis

(16)

According

to

equation

(16}

the

ooefficient

o~ of a ter.rn of the radial distribution function snould be

proportional to the height ot the corresponding term

of the representation o:t the

x.·edueed

intensity :function,

to ak and to the square root ot b~ • It may be inferred from the detailed treatment of the example

ot a rigid diatomic molecule that the best value of

bH 1s g1ven by t>K ~0.5'4 cs~-s"'lwhere <s.:--s .. > is the

halt-width of the k-th feature of the reduced int;ensi ty

curve.

Finally, 1 t can be seen from the form of the

reduoed intensity funotion (15) that tor a rigid

molecule {the distribution

net')

is discrete) 1 t

and equations (8) and (16) follows the impor"tant

corollary that on the average the terms of' the radial distribution function (8) for a rigid molecule are

te:rnpe ; its

to t uee an only of ter:n:s

in the radial distribution sUDJIUation.

A:pplioation

ot

the results so tar obtained to observed diffraction data does not yield thoroughly satisfaotory radial distribution functions. For

example, Degard found that even when oarefully

deter.mined intensities are used, the radial

distribu-tion function calculated for carbon tetrachloride in this way shows many relatively -weak, false maxima in

addi t1on to those oorrespond1ng to the a-Cl and Cl-Cl

distanoes in the molecule. Dr. Degard found that these

a~urious features oan be eliminated by using in the

radial distribution function instead of the t:rue

a. s"

in which a intensities I(s), quantities I{s) e

_..:z s

-is chosen so that

e

=

l/lO where s ... 1s the upper limit of the integration or su:m:mation~ The form of the

-a..s•

expression e suggests the nrun.e "artificial

15

by j;J.st such a factor.. 'I1he artificial temperature

factor rapidly and progressively dec1~eases the b1portance

ot

the _{of the distribution function corresponding}

s of s ; its action is _{that of}

t _{a smooth compromise be the tl:leoretioal}

demand parts

or

the radial distribution arising

from all values of s should on the be equa.lly

in1portant, and the practical necessity of giving s a. f1n1 te upper limit corresponding to the greatest observed.

angle of scattering~

'.Phe ett'eot ot the e.rtifioial temperature factor on tl1e significant peaks of the rad.'ie.l distribution function

oa.n be 1cipated from the ciroumst:ance mentioned in the

last paragraph--that this factor is of the same form as

the factor which occurs in the theoretical intensity

function tor a vibrating molecule. A distribution function involving the artificial temperature factor corresponds

to a model in ~vh1ch every 1nteratom1o distance is sub jeot

to the actual mean square variation in the molecule, plus that corresponding to the artificial temperature

_ ,

fac·tor,

~

Le.,

f,,e;/=

.J4·+za

2• Consequently, if it is desired

l. When this is done, the terms col'responding to large s values beoome unimportant, and the extension of the

upper limit of variability of s to infinity, made in the derivation of the radial distribution formula, is

justified.

16

to determine the a.mpli tude of the moleouler vibrations

from the width

or

a peak of the radial distr1but1on

t it is necessary to t a term

correspond-ific

1 ial tor

valid. if the arti:t'ic ial temperature :f'ac

chosen large enough so ttiat the real

d by

i O!J

has been

i1'1ojal

temperature f'aotors together render unimportant the

parts of the radial distribution function corresponding

to s > s,,,., which have been negleo d, it the

representing the distance in question is well separated

from its neighbors.

Degard has gi van a different, more detailed.

t1•eat-ment1 of this question based on an approximate evaluation

ot

the 1noomplete radial distribution function

-(17)

I(s) has been given

the

fo:mn,

(18)

ap:pl1oable

to a non•rigid diatomic molecule, and suitable when an

artificial temperature factor is used. Degard1_{s 1ntegra~}

tion is valid only tor (.-e'-:::.£'1 )-:::t...O, so that it can only

l. ~hesis, Chapter 8. Note that in this paragraph b

17.

give information about the near-maximal region of the

rn.a1n pe or D' (~}. Degard has calculated the half

th of this peak at thl'ee-gµp.rt§lr.-h!igh,t as a

tu.notion of and

of this :peak, as a funot ion of 11. &nd s..,,~ 'o 1s

large and very nearly independent of a for small values of ilm ; for larger values of' s,,., , b decreases rapidly,

and asymtotioally approaches the value a.. .And the

the value of a, the smeller is the value

ot

s,.,.,

for

b~a. Unfortunately

Degard's curves seem

to be aub ject to err·or, for certain theore·tical re lat ions among them are not precisely satisfied.

For the case when s,.,._-.;i..oc::» the value

or

the integral

is exactly known and the resulting function1 is at importance

here, and is

or

great interest 1n connection with the

theory of the temperature effect inasmuch as it gives :i;irecisely t.he error incurred in omitting the highex- order

terms ot that theor1, and suggests a oorreotion which is

almost exact. However, this correction will not be dis-cussed in this thesis,

The substitution of (lB) in (19) gives

18 ..

whioh, _{s rn ~ - - ' is found to}

(20)

( _{is the}

distribution of scattering matter oorresponding to

(18), with essentially the norm ( £PCP/d7f

=' }

and

the half-width

{tu;-:::::.

a, in formulating (18)'

the faeto:r

__.ea

being unimportant in these respects .. This tor is effective, however, in that it shifts

the peak ot the radial distribution function so that

the art1r1c ial temperatu1·e facto1· gives

rise to error, which, however, may be eliminated by

taking, as the interatomio distances given by the radial

distributio:n method, the peaks

ot_t?flKe/=LJCP,/Lf'

rather than those

ot

D(---f) • The :re&l temperature :factor

corresponding to the actual intensities, differs from _aslt

It seemed desirable for the present thesis to

supplement Degard's discussion of the effect of a

tini te s,.,, , and of the temperature faotor, with the

furtller oonside ions. It is impossible,

been ioa , ·to a simple

expres-sion of' the 11_1neomplete11 _{radial distri'bution ion}

for the oase wllen I ( s) ino ludes a temperature :f'aotor.

IIO\"leve:r:, for our :purposes, /-1'-_z ~ c.s oan be made to

-CL.Sa,

be a sufficiently good approximation

ot

e , as is

illustrated by Figure II. We shall investigate the

nature of the radial distribution function for a.

diatomic molecule when o is given various values.

For the comparison of tbe results of this investigation

with the corresponding radial distribution functions

involving the

oorreet form of the temperature factor,

_a..s"" it is necessary to find what value of a makes e.

/+~cs

and ;e most nearly the same for O ~ s :!, s,.... •

In

Figurfa II

where

a =-0.28 o2, {b -=-0.0l; o =-0.169),

the two for.ms of the temperature faotor have been made

to have the value

t

tor the same value of s. This

relation

of o to a is seen to give a good

represents-tion of the true temperature factor over the range

- ... s ..

I>,. e

..

For the range / >.- e ? c!J. t

better average representation is obtained if

' a

2

20 •

.i:\.nd it 1 t is desired to obtain a representation over a still larger range of the temperature factor, a must

be given a larger value st 1

theoretical radial distribution funct for a dititomio having the temperature factor discussed in the las·t paragraph can be set u:p and

evaluated straightforwardly. We have

, and hence,

(23)

In

order that this function shall correspond to integration over a definite num.ber of maxima and minim.a of the reduced intensity function, we $et

(24) _• Then,

21~

On making the t'ollo·vving subst 1 tut ions

simplifying,

as

:ror a d1atom1o molecule and the a p:proxi:ma te temper el ture factor I+ CO$C'$'

2 This result represents n features

(maxima and minima being counted separately} of the

reduoed intensity curve. In the absence of a temperature

factor (o=O), (27) beootiies muoh simpler:

>t

( 28) /)(€/ ::; :.l(-J) ~ •

..f''L

:O(..t)

=

..l2

D'{.€) has been evaluated for n=5, 10,

and

:ao

and C = O,

l/n,

and, for n

=

10~ 0.8/n. The

results are plotted to an arbitrary saale in Figure III

~

where the convenient notation D'/ (-€} has been adopted

to represent D( L ) for n

=

x and O

=

y. It should be noted that C

=

O corresponds to a unit temperature faotor;

O

=

o.a/n:t

to a factor equal to O.l when s - s_ ; and

0 - l/n, to an approximate faotor equal to

zero

when

'"" em • The last ease is the extreme li:m.1 t

ot

the

-a.s•

usefulness of the a:p:proximation of e by 1 +coses

2

22.

which has only a very small value ( 0 .. 03) \Vhen

J .. part of the peak of Dt( .L

L

and

ot

.ID'( ,f ), tor each choioe n and Cl is also shown in Figure IIL Bi.nee the height of the main peek

1

D { .f ) is pro port 1ona 1

by dividing by

n,

1 t to

n, and this has been done~

These theoretioal curves are of interest in

oon-neotion w:tth radial distribution functions in general, because for any moleoule the distribution function may be regarded as a superposition of several curves of this· type. radial distribution funotion will be

satis-factory if each of the component parts has the form of a peak with level plains at either side, and if the various peaks are not too closely s:paoed to be well

resolved. luid tllis will be true also in the oaee where

the radial distribution function is calculated as the sum

ot

several 1',ourier tern~s, with suitable eoeftia 1ents,

if the conditions necessary tor this approximation, as

discussed in section ~2, have been satisfied.

'rhe features of the curves, of Figure III, which

appear to be specially important will now be discussed. 'l'hose obtained w1 thout a temperature factor are seen to

l. This is easily seen, from (28) for instance, by

evc1l~ating the indeterminant expression D' (t== l)

have a oentral important peak and ne igh'boring lesser

peaks which are progressively smaller, the further they

are _{the main peak.} :L1_{he nodes are equally} spaoed except at tlle main peak, which, its nodes,

is twice a.s Vii de

es

the other features.

ions _{D9,, (.t)} i1 have main as

as those for which C 0, while ·the

ot

the curve

the L very closely, the corresponding minor features of' the f'irst curves having almost entirely

disappeared. The function ( -t ) has a somewhat

""'""''""' ... ,,..""""' value, while ·the spu1·ious features ni:rve

become negligible for it also1• This is the desired

effect, for the superfluous features ot the radial

distribution function 1n the praotical case, when no

temperature factor is employed, are very confusing and

render its interpretation very difficult. It may be

well to remark that probably any other function which

suitably reduces to very small importance parts of the

radial distribution SUlll or integral corresponding to the

outer rings of the d iff'raotion photograph, would serve

the same purpose e.s the artit'ioial temperature factor.

It may also be worthw11e to mention ag81n tnat the

artificial and real temperature taotors act iu exactly

l. Perhaps a less drastic factor still would serve.

How-ever, th.is one, chosen to have the value 1/10 for s =Sm,

according to the suggestion of Degard, is seen to be sat1sfaotory, and is perhaps &s setisfaotory as any

the same way, so that as t _{real temperature factor}

beoomes more important, the artificial te11perature

become less so,,

width of the peak of' the redial distribution

function is of i:nte , tor 1't de the resolution Whicll :l.t is :possible to obtain when several of the

silnple curves (for different

1'

~s} are added together 1n the praot1oa1 case. 'l'he half-widths at the :pee.ks

ot

the curves ot Figure I I I ere found to be 0.6/n

tor

C ; O; 1.0/n ·_f' for C -=l/n; and, presumably,

.o.e/n

·.£.'for

o

o.a/n.

l 'l'hat these reaults for the

cases C

=

l/n and C = O. 8/n are in agreement with the

general statements concerning the width of a radial

distribution peak given earlier in this section ( B16),

oan be seen :trom the following table which gives the

balf-wid th of the peak of the radial distribution ourve, the

value of a (a=

o.

30 o2 } corresponding to C, and the

half-width

ot

the distribution

or

scattering matter

oor--a~2 G.'A~

respondinig to the temperat1lre factor e ( .cij = 2a;

for a, Gaussian distribution

X~

=

1.1?6

ff. ) ·

l.. Tl:l.e generalization tor C ""0.6/n is justified by the

apparent circumstanoe that the various curves tor a given value of C (e.g. l/n) are the same with regard to general appearance and width of the peak exoept tor the scale of (L-l). That is, they are similar

Observed Half-width

oal-Ourve half-width _{a culated. from a}

$'

D9:s ·O 0 018 0.193

'"

D~ .. 0 .. 10 0.,003 0 .. 091

n""

'124 _"

o ..

0 .. 046

D'" _9-11/" 0 06

o.

0~0

For general radial distribution functions the value

ot

n to be used in estimating the expeoted half•width

ot

a particular peak is tlle nu.mber ot maxima and minima of the corresponding component of the r~duoed intensity :f'unction virnioh lie between the origin and a"' • It is convenient to state the above results in another form

tor :praotioal use. This is easily done by noting that

(29)

,

Swr..fl

n,·

=

7( and making use of the definition

ot

a,

(26). The results are given in the following table.

Theoretical Half•widths of Peaks of the Inoom:plete Radial Distribution Function with a Temperature Factor

c 1'em:perature Factor Half-width

{corresponding}

0 l l 1.9/sm

8.51/am

O.l O.l 2.5/sm

Two peaks of equal height may be expected to be well resolved 1f their separation be greater than three times the ave:rage half-'ivid·th; or, resolved, but shif·ted

eaoh other it' their separation be twice

their width. 1c·t o:r "'Gb1s table

are in general agreement with experience tor radial

distribution curves oaloulated by the modified method described in seotion

o.

This is true with regard to

the magnitudes of halt-widths as well as their equality

tor all the well resolved peaks, having like real

temperature factors, of any on.e redial distribution

function.

Finally we direct our attention to the positions of the maxima of the curves of Figure III. It is seen that in general the maxima of D(..l) ooour tor L slightly greater than the theoretical value of unity, while the m~xima of D'(..i) occur when L is slightly less than one,

and those of _,f

nt

(1 ) are at t

=

l. That this should be true is evident from the fo:rm of ( 2?) tor

o:::

O, and

by (21), tor any ohoice of the usual temperature factor, also if em be large enough. From (2?) it is evident

27.

megni tudes of the shifts cannot be s1luply expressed. The values of these slli:tts as given in the following

table are obviously only approximate; better ones could

best be obtained by a more precise oaloulation _{{ 27),}

:iinoe the analytical t;reatmer1.t of ( 27} 1s eLJ;pa:r·ently

very tedious.

Shifts of Ma x:tm.e.

of Inoo:mplete Radial Distribution Functions

Bb.1:f't ot IVrax1mum ot

n

c

D (.f ) _D' (.f )

5 0 -I

o ..

007

-o.oao

lO 0 + 0.004 -0.004

20 0

___

...

---~-(very small)

5 1/5

.,. o.

025 _-0.033

lO l/10 + 0.007 -0.006

ao

l/20 ..,. 0.002 -0.002

10

o.a/10

.,. 0. 00 7 -0.006

The error depending on lh1s etfect can probably be

eliminated in general by taking the positions of the

maxima of ..ID' (..i' } instead of those of either D ( .t }

or D t {..I } .. In general, it mJ'.J.y be e::cpeoted that these

l!lhifts may vary from about a per cent tor n

=

6, to about a tenth of e. per cent for n = 20, and to have the sign

B. Applications

method of Pauling Brockway; the i tion ·to a modified metb.od.

The redial distribution fo1'T:1ula as derived by

was simplified by them to the

ct io al use as er! aj.d in the

determina'tion of molecular structures of gas moleoules

by electron diffraction.

(

D /. , -

\ I

9i1tSt-€

f1J -

L

·J.

s

..1

-/( .,.. 'k. '

Here Ik is the visually estimated 1ntensi t~r of the h

ring the electron d1fi'raot1on pho1'ograph, and. the

other symbols are already familiar. This function has

been applied in the study or many moleaulee and has enjoyed success, especially

ror

molecules with very

few, and well separated, intaratomic distances. #hen

these conditions are met, the function yieids curves

whose maxima oocur at values of

1

in good agreement with those obtained by the visual method of interpreta ....

tion. There are slight disarepenoies among the inter ...

atom::Lo distances found by (30), however, and it will be attempted to discuss tbese later. But great

dif-f j cul ty is met wl1e1n the molecule under consideration

involves a considerable number of significant interatomio

distances, 1'or the peaks ob·tained from {.30) are

excessively broad, and consequently the 11reso1v1ng

is poor.

of this

a suggestion ial

die-tri'bution me·t could be

treatlllent of the ooetf'ioients of the s1n sk..I terms

of the sum of equation (30)

distr"ibution functions calculated trom the observed

posi·tions and visually estimated intensities of the

of electron diffraction photographs of thiophene

by an expression similar to (3D) in which the aoef•

fiaients ot (~O) have been replaced by lks -1s

~ ~ +

Ik , Iks , lks , !ks , Ika , in A, B, 0) D,

E~

and F,

respectively. Inasmuch as the data are reported in

part two ot th1s thesis 1t will be unneoessery to give

them here also. The interatomio distances in thiophene,

as finally determined jointly by the visual and radial

distribution :methods, are indicated by arrows the lengths

ot

whioh are proportional to the number and atom1o-number

produot of the corresponding interatomio distances. It

is seen that e.s inorea.sing weight is given to the outer

rings, by inoreasing the exponent of sk in the

30

There appear several significant pealts which are

either poorly resolved, or not resolved at all, in

the first tnree curves, and these ooi-respond well

with corr·espo:nd ing inte:rerto:mio dis oes. It is

e ly s ·that "Che posi·tion~ the

greater peed:(S are ted only slightly by

ve.rying the treati:nent of' the ooettioients. This is

important in that it shows how the neoessarily large

uncert~int1es of the trend or scale of visually

estimated intensities oannot give rise to great

errors in the determination of interatomio distance

by the redial distribution method, and how both the older method and the modified m.ethod presented here can give good :results. It is also to be notioed that

in E and F there ere many more maxima than in , B, and

o,

and that some

or

these new peaks appear to have

no physical signifioanoe.

A similar set of radial distribution functions

was calculated from the data tor benzene with exaotly

similar results. The peak oorresponding to tbe distance

between carbon atoms at opposite aides of the benzene

ring e,ppea.rs at the oorreo t plaoe al though 1 t was

2. :t'he ified Visual Radial Distribution

Method~

on visual estimates ot intensities,

for various molecules, made by workers in this

labor-t _{correct simplified} t t al eu1·ves

these :rm.;;ileoule~, indicates t t the visual inten::ii

is approximately proportional to (l/s) times the

simplified" theoretioal intensity, or (s ) times the

theoret:Loal intensity. It was therefore suggested that

the

visual

radi&l distribution function should be

given

t form

L

~ N. fl

(31)

Dt.t)=

L~~e-a,,s,.

:;,,.,,;s,)

where I(s}

'*"'

_J

in { 7) has been replaoed by Ik<wl>·J ( s )/s;:;, the factor ₁₂<sJ

has been left out, the integral has been replaced by a

summation as in the older method, Degard's artificial

temperature factor, in whioh it is customary to choose

-as2

a suoh that e ,..,

=

1/10, has been :1.nserted, and the

sUllllllat1on extends over all the peaks of the observed

1ntens1 ty ou1·ve e:x:oept that ~'b e .-:. 0, the terms cor-responding to the first two features of the reduced

theoretioal ourve being thus omitted.

Curves calculated by equation (31) are found to

"

and in the similar oaloula. t ions for benzene. Moreover

the prominence of tl:1e spurious maxima charaoteristio

E and F 1s in general greatly reduced as a result

of the action the artifio ,temperature tor~

of' s t.besis aeve ere given 1

I1hrough the efforts of fri

s o:f' ( 31)

111 t s

, for whose special assistance in this regard

the imthor is most grateful, 1 the new func"!lion has been

tested for many othe1~ molecules of di verse types, and

has been found to be :powe:t•ful and generally reliable.

It has become especially evid.ent that even rnhen the

function is not completely satisfactory, in that the

interatomio distar:.ces obtained are not entirely

compatible with a reasonable molecular model, it is

nevertheless true that this s't of interatomic distances,

together with suitable we1 gh·t faotors, does often lead

to a simplif'ied theoretical curve in complete agreement

w1 th the appearance of tlle original data, so that the radial distribution method leads to a solution of the

problem which, if not the correct solution, is at least

indl.st1.ngu1shable from 1 t on the grounds of' the electron

ditfraotion data alone. There persists some difficulty

about spurious maxima. These often can be recognized

al though perhaps :rela ·t 1 vely prominent to the eye,

have a maximum value of D' (

.P )

flpproximately

zero. tive values of' D'{j?) probably arise from the omission of the f'irst _{terms of' the}

function (and are hence _{). while spurious}

al"e to

imperfeotions of t function. discus,sion

the modified method 1s given section 05d.

3 The modified radial distribution method

discussed in the last section is related to tl1e general

theoretioal radial distribution function by a series

of app:i::·oximations, some of which have been justified in the theory given in part B, While others must now be disoussed s:peoial1y. To the first elass belong the substitution of summation for integration, and the use

of an only finite number of terms. To the second class

.J

belong the omission of the factor .fat:» , the omission

ot the part of the integral or summ.ation corresponding

to the first terms o:f' the reduced 1ntensi ty function,

-b1e.t ...

the omission of the factors e demanded by the theory for the substitution ot a surritllation for

integra-tion, the interpretation of D' (.I ), rather than D ( R ) ,

as giving the importance and values of the 1nteratom1c

distances,

artificial

the effect of these things and of the

ternperature factor on the positions of the

discussed

of intemsi

t omitt

the su:mrnation; the

f noted

ta,

the first terms J.

first features of the reduced theoretical curve is ne

1.

equivalent to the omission of f'!lfs) For, as is illustrated by figure Tf, this taotor is nearly constant except small values of s, for which it becomes

very large, approaching infinity as s approaches zero ..

Consequently the first terms of the radial distribution

function, which, ·..vi th the factor, are severally of

a:p-:pro:xim.ately the same importance as each of the other ter:rus, ·become wi.thout it entirely negligible. 1_.I1_{he se}

parts ot tl1e radial distr1but1on function are essentially

:positive for the range of values of ..R. which are of

interest, so that a function from which they have been

omitted will be depreaaed in this region and will show

negative values of D • ( f ) . Professor Pauling nas pointed

out that such a radial distribution function, in which

the a.tom form factors have not been considered, is a

representation of the actual distribution of scattering

matter the molecule, the electrons having a negative

35.

l

of Degard are in agr.eement with these general

statements. _{found that radial distribution functions}

for oarbon teti'aohlor1de and bromine calculated from

accurately measured intensities according to the methods

ocutlin.ed abOil6 a.re eYerywhere essentially posltiVe if the tor is 1naluded1 but have a ctable minima

ad jecent ~to the peaks i:f 1 t is disregarded Perhe:pe

t l:le of' the moa1f1ed rallial Cl1str1but1on functions

are partly due to this situation.

J

Since the function /'llfSJ is not :really constant

beyond a 'l:lfi' 5, nor is the part of radial distribution

function from small va1ues ot s independent of

,f ,

it is desirable to consider these points in greater

detail, even if it is not possible to demonstrate

oon-olusively that radial distribution functions constructed

from visually observed data have anything to do with

them. While a suggestion regarding the treatment of

J

the lesser variability of

.F,

_(S)

{

s

>

5) is made in the

la.st section of this discussion, we cannot expect that

'the estimates of intensity, as they are now made, can

be

as

oonsistent with regard to scale, over a large range

of $, as would be necessary to make a consideration of

J.

fa<SJ ,

in this region significant. An attempt to estimate the etfeot on positions of peaks of omitting the first

terme of tb.e radial distribution function is given in 030.

b ..

2 -6..f2

The f'a:o tor , /

e

..

The modified radial distribution :runction

described in section 2 :m.ust ao

'

to ·the theory,

be d ed by the

be e cted to

a:reas correspond to the probability{ multiplied by

the product of the atomic numbers) of finding in the

molecule two atoms a distance

1

a:part. In the

ap-plications to molecules, using visually estimated intensities, 'both these :rectors have always been

neglected, and indeed 1 t has always been assumed that

D' { .f} is the function of :physical interest. We shall

see, however, that these co+reotions are sign.if:toant.

It turns out thet the physically insignificant peaks of the distribution functions which ooour tor both

relatively small and relatively large values ot

_I

axe

ellln.inated by these corrections, and do not arise from inherent errors of the radial distribution method. (These

pe~ks are already oonsiderably weakened in the transition

trom the old to tbe modified method).

If it is assumed that even when the features of the intensity tunction are not all ot the same half-width,

of obtained from the average width of the features

of the inte.nsi ty function, then all of the exponential

functions at

1

become the same and the corrections

discussed in this paragraph beoome s.~,••J•.1 a fao by

whioh the distribution ' D1(..f)

is to be multiplied~ This is very desirable it

greatly simplifies the ealaulation the corrected

modified function~ And certainly the use of an average

ia suitable for the pre~ent :purpose, but it

must be understood th~t the use of such a factor does

not i1uprove the representation of ·the reduoed theox·etioa:L intens1·ty function upon which the substitution of a

sUll'.IID.8tion for integration depends.

/)2 - b.itl.

The faotor Ff.P) ""..x e has been plotted in Figu:re VI as a function of s f'or a range of values

01· 'b. The table in this :figu.re gives the value ot b,

and of the oorresponding average values

ot

half-width,

(sk'~

sk), and 1nteratom1o distance,

-1

1

, for eaoh

curve. The two values of

.1'

are obtained from the

..J

i:. 71'

1'-

7(

express ions - S<<S,f-.&,> and - ₂

ts;,-

sJ

for the respective oases when maxima and minima, and maxima

only are used. The "average" reduced intensity function

is taken as a sim.ple sine curve, and b is obtained from

2

the expression of section B2, b

=

0.5 (sk'- sk) • Since

:38.

on the same scale it is well to note that they are

inter-related simply:

..

,

It is easily ~ t t _ch

ie :t'elate t::-1 the ters b,

(a~'- ) _and

..t'

_{by the following expression:}

(33)

_fm

= -J

=

t.4

=

{ 0.9.f'

(maxima only)

~

(S' - :,• )

1.35..t'

(maxima and minima).

k A:

With the help of Figure VI, and the eoove auxiliary

expressions, it is now easy either to correct the modified radial distribution funatio:n before plotting

it, or to estimate the relative importance of its peaks if it is plotted without oorreotion. Obviously

the uncertainty in the semi ... quantitative for that reason.

o. Shifts of peaks

or

the radial distribution

function.

It is evident that the corrections disoussed

under the last two headings imply that the peaks o:f'

D'(L} will not in general give the oorrect 1nteratom1o

distances, but will be shifted somewhat instead. In this section an attempt ia made to estimate the shift

quantitatively, and it is found that the shifts are

39

the widths _{the peaks of the radial dis tri.but ion} ourve. Generally, the effect is eomewhat, to spread

apart the of' D (

l ) .

1I1

lle predicted shi s are

too to be oonfirmed, exoe _rh tatively,

1 dis ions oaloule d tne rnod if'ied

method from :reasonably complete date.$ func·tions

oaloula ted by the older method shov1 greater e:r1•ore,

an effort is :made to test the 1_{predictions 011 them. r1}1_he

application of the theory to the results of the older

method is reasonable, because the older method corresponds,

at least roughly, to the use of a vary drastic tem},iera ture

factor. A peak of the theoretical R.D. function for

an interatomio distance having a Gaussian distribution.

about a mean f,· , with half-width Wl is gi van by (~ -..£.·)2

) D ')

-

L.41-W2

( 34 lf-1!. =

e

and peaks of n1ncomplete" radial distribution functions

are of this form also, if the temperature factor be suf-ficiently im:po1·tant. We assume, therefore, that in

general it is suitable to represent the peaks of a

radial distribution function, D(..t), by (34), and if

a peak hes not the ideal form an attempt is made to

choose W1 so that the top of the :peak is well represented.

If W; is reasonably. small, the peaks of the modified

form and width as ·n1ose ot' D; (.,/, ) , and W may be

determined directly rrom them. now write

(.€--1,·Ja

Le -

1..44-wfl

~pt: -h-€JL

to the detini tion of

' ( .i )

and ·the

foregoing discussions, p must equal

2

in order that the various :peaks of DJn ( ~ ) as defined by ( 35} s)lall

have the same relative importa.noe as 1n D'm(.l) as

actually oalculated by (31). On the other hand, according

to section B3, p : l ; , i t 1 't is des1:red that the m (

L }

oaloulated from {35) and

(3l)

give the same interatomic distances.

Now i f the peu1cs of D(.l ) are well resolved, the

position of the 1nax1mum of any one can be obtained while

the others are ignored. Accordingly, we consider

(L --1.;J'

e

.1.#w"'

...fp e-11.t2

< 36)

D'

_me .<£J = for which

b.la- U--liJa.

~n' IL') J.Hw..,, { -<,o-1J ,,,,,_ "' .. ·• -p -Ip"'']

( r:t 7) Cl!' Vwt/ =

e

2b JI , '7 r.c

..c.""

A , IJ

"" d..JZ ..x. - ;/(. 1.'41/ wa A , -

p

..£

( 3?

l

has zeros at _/-::; o , _.,l;:,: L.t" ,

t

:= ao • The

41.

p+L

when multiplied by ....f. and expressed as t1o ( .£-.£« ). Sinoe the roots of this

quadrat ion are widely separated in absolute

approximate solution,

-

-is auf:C1c:!.ently accurate. {For cases of :Ln terest, the

last express ion of ( ..i--L1 ) may be in error by 10~~. max

In the following table are given the celoulatea errors tor the radial distribution functions for a number of

l

molecules calculated by Pauling and Brockway a.ocording to their method, together with the observed errors {based

on the structures) determined by the visual method. b has

been estimated from Figure VI ("maxima only") and a

weighted average distance

...f' ,

while

w

is estimated from the radial distribution curves. That it has been

necessary to choose p 2 (instead of l) to obtain the

following remar~able agreement is difficult to explain,

except as evidence for the very rough character of this

discussion. r~olecules having only one interatomic distance

have been omitted.

M. -

x

- .A v

max.

Molecule ,£' b .&V

J;

Mcalc., Mebs. tP

fo

~4...alc., Al'abs.

CF 1 .. 9 .35 .. 28 L.3.35 - .. 031 -.015 ., [email protected] +.010

4 ~4'

CiF 2.0 .. JO .26 1.5.27 -.019 -.017 ?'"' .-. + _+.0.30

.., 4 _""'"'' "'•

cc1

4

2 .. 5 .. 20 .25 1.74 -.020 -.020 .20

r,

.+,Oll -.,010

,~,.,

SnC1₁ 2.7 .. 16 I _{1.963 -.OM. -.!'l53 -.016}

•"+

....

Gec1

4

2.5 .. 20 .3 2 .. 070 - .. 009 -~033 •.J -:i J.3135 -.055 Snc1

4 2*7 .. 16 .27 2 .. 2:;15 -.007 +.006

.

.,) 'J 3.76 +.030 +.015

PF~ l .. S .. ~.O .. 22 1 .. 47 -.006 -~050 .22 ;::. " +. +.040

_,

PC~ 2.4 .. 22 .22 l .. 9B - .. 005 -.,040 • .3 3.08 +,,044 - .. 010

ff:>.

AsC1

3 2.4 • .22 .31 2 .. 13 0 -.050 .37 3 +" +.070

N

(H2)

cc1,,,

2.3 .24 .24 (L 73) -.013 - .. 040

.

21;:: _, .c. .. ,..., + • +.015

...

(H)

cc1

3 2.5 .20 .16 (l..77) -,,000 - .. 010 .31 .2$9Jl + +.001

ClriO 2 .. 1 .27 .27 l.65 -.016 -.006 '") _{+ +.004}

• .;:.,,,

43.

part of the radial distribution function

on the first features or

:maximura and 111int?m.uu) of the reduced intensity function is t _{form - - - - or· - - - - -} the

t·No cases men l;ion<;;id, i ~s maxi.111U111 be igJ:rt vary1ug fI'Oill

2/n - 4./n times ·~he height main peal:u:1 o1' tlle

distribution function, according to the importance of

te1nperature :factor. Omission of this term will

lead ta a spreading apart of the peaks of muah the same

nature as that discussed in the last pa.ragraphs. 2.'he

shifts due to this eause will deareas~ with increasing

sm for t·flo reasons-... the narrowing the peaks and the deoreasing relative importance of the omitted terms.

It does

not

seem worthwhile to give a quantitati've dis-cussion of this effeot: the last example illustrates the approximate magnitude of this one also, and suggests

how either oan be discussed.

4. Disoussion and Suggested further :£',Iodifioations.

The study of the radial distribution function for use with visually observed data is not complete. In

order to gain a thoroughly satisfactory understanding

accuraoy of the representation the reduced intensity

f'unction, h is neoessary, o:r obtainable, for the

various si ions arise. eff'ioaoy of

-tl.e-<'.e

choosing the appropriate values of e :t'or

dif'-terrns the ion should be investigated,

as 1s speoi:f'ioally dem.anded by the theory given

this id done the explicit form the

coett101ents, given in B2, should be tollowed, and it

would perhaps be well to include in the radial

distribu-tion funcdistribu-tion, terms tor the first maximum and minimum

of the reduced internsi ty function Of ooui·.s, good

measu1·em.ents of these features cannot -i:ie had but they

have to be known only so approximately that this n.eed not oause trouble.

Two suggestions which certainly are of pr~otical

importance

oen

be made. First, it has been found useful in the few examples which have been tried, some of whioh

are included in Part II, to 1nolude terms for both maxima and minima according to the suggestion of

Dr. Simon Bauer. Although it is somewhat difficult to estimate the negative ooeffioients for these terms,

and i.t is possibly true that visual measurements o:f t1minim:an are not as reliable as visual measurements of maxima, these terms do make a reliable and sometimes

function,

spurious pe

snould therefore be usea. Often the

of the function f'rom the m.inima tend

to oancel those of the maxima. great difficulty

in the visual estimation of intensities is in

estab-trend. s can be

avoided by arbitrarily fixing tihe scale of the n1nte:as

estimates in suoh a. way that on the average they are oonstant--they are to represent the a.mpl1 tude of the

reduoed theoretioal intensity function for a rigid

moleoule. By thus estimating the aoeffioients of ·the

reduced intensity function the J factor is

nioely provided for, while the average real temperature

faotor for the moleeule is reduced to unity, so that

differences among the real temperature tactore

oor-respond ing to the several interatomio distances will

be left unchanged. ·1v1th these intensity estimates the ooetfioiente beoome

(40) where

is the usual artifioial temperature factor.

It may be expected that the re.dial distribution

method which has been described is as reliable as the

visual :method (and much :more reliable and powerful than

oases in ~ radial distribution fur1otion from

visually observed data must always be inferior.. Suoh

are tne examples wr1are many of' the i:rnpor·ta.nt features of the intensity !'unction are for some reason unsuited

to quent ive ·treatment. i.f'hese t'ee:tures can be

avoided, as such, in the visual method of in~erpretation,

but in t

equally

radial distribution method. they must be used

the rest of the de.ta. Perhaps the re.sul ts

of the visual radial distribution method r:.aust still be

regarded with awe and :f'aith, even though the theoretical

expressions t11e halt-width (der1ved 1n ) indioe,.te

the expected resolution; the curves of Figure VI help

to tell whether a peak at a. relatively small or relatively

large value of

-i.

is significant, and the work of Degerd

has provided a clear just1f1oation for the in·trepid

,] ~J

t '

vaJ:uet:J

:t\:mction :forrmilated orif;inally :for single bond-double bond resommce of the carbon-carbon bond (l ). '.!!here a:r;a giilen in this section the results of the determination

conclusions regardine thfili:r electronic str'llct'\U'es :reached 1:Jy application o:i' the bond t;ype - intera.to.mic distance f'llJlCtion. In addition, beca:r.ise o:f the 1.mprov~er.tt 1n electron d1i'!ract1on technique since tr ... e st~ of benzene in ·these Laboratories four years a.go (2) and because of its similarity ·~o pyridine and pyrazine, this substance mw re-investigated.

~!·~a~Ps!·- T'he diffraction photographs were prepared with the apparatus and technique described by :Brockwa\V (3). Ten or roox-e photogrEJqJhB

were made :for each substance. the electron wave length used be~.ng about

m~whri.ng ootipariao:n m:ioh Y.i.1rup1ifi.ed the or et ioal intensity tlW''Vtlll

fer the models 1.tlider iiscusuion.

'bt::n2ene and. :t'ejec'ting the :first and la.st portions. A similar trea:tment was gi.ven to pyridine (Xahlba~), p;r:rrole (East1aar.1.),. and thiophene (Ea.strran).

Tlith gra.n.·ulated zinc and :rediatilling the prodi.i.ct several ·times. Oyclo-ptm.taaie:ne was :prepared by d.istillL~ dicyolopentadiene; in order that

di:ff:raotion photographs wez·e made i:rrmediately after the distillatii.in (within one hc1"Ul'). F-1..xra:n was :prepared by the deca:rboJS\)"lat ion of' :t'woic acid a:od purified by :fractional distillation. A saiqple of py:razinfl kindly given us

'by Pro:f'as~wr F.

w.

:Bergstrom was used without :fu:rthe:r 9~r •.ficatio:~.

}.!..ep~~~₁...llr .. idlp.f!.t.-~

ll.fM..!11...!·-

New photographs of banzen.e were obtained showing four more mea.au.:rable ::ri:ngs than those used in the ea:dier investi-gatio.n in the1.ui La.baratori~m. The radial diotribution C'\l.l'Ve oalc'Ula.ted

'ffith. the o and s

-

-

values of Table I (Fig. 1) shows three principal peeks,

0 0 0 Q Q

the

o:f the

designated by

it to be a or in:ner and

ma:tima Vil'Ual

and f ow th ri:Dgs have _4/3;

0

C-H = 1.12 A or more. which :u:eke the :fourth riDg as as the .the 1*03 cwve (as well as the

shows only a :poorly de:f ined shelf in place The

the

0

0-H

=

075

A (the mean 1.06 and 09 Ci.U'ves) Table I, leads to

1lll

• 39} 0.02 A

0

0-li = l. OS .:!: O. 04 A •

Wierl (4) a.nd Pai.tl:i.ng and :B:rockwa¥ (2)

0

C-0 = l .. 39 A in (5)

c-c •

4o

.±

0.01

value is in error.

pyridine {Fig. 1)

311Cl C-l:I

=

08 P.

<fl

dis·ta;nca L3.g5 0.01 A. The cc1u;pa;rison mt'.h t!rn ..! val·uea a

,,

l"eg'.:ilax· hexagonal :model leads to the aam:;,i "la1'1e 1.365 A :f ru· the a"'n.,:cage ring ·bond distance-. It seams Utl.l:Ucely that. the C-C di-eita:nce diffa.t•.a

Ill

£':.~ma 1.39 A, and the :re:c:u.l:t!I acco:t·d.:Uigly t:ruggesc ·that the C·-~a :Ohtan;:;~:;i !I

about 1. 37

..±

O" 1)2 A.

The principal peaks on ·the radial di~1trib'11tio.n C'\ll'Ve fo1· py:t'azi.l'Hl (Fig. 1) li~ at :1..36, 2.37 (

""'1}.

i.37), and 2.76 ( .. 2 • i.3e) ~.

~ <ll

Intensity C1ll'Yas :fc:.· mod.el !, wj:th 0-0 = 1 .. 39 A , C-N

=

1.33 A , C-H =

•

1:.09 A , and all ring angles 120"',, and fox· 1oodel II, with the same bt1nd

malcing the meta distances eql:lal • are shown in JJ'igure 3. These C'Ul'"Tes,

as well a.a the benzene c:iirve (nipresenting a regular r1e:xagona1 model),

agree qiialitat:l:veljr with the photographs. The qu.a.ut.itative corapa1·ison :for

'Cl

all th:ree lead.a ·to ·the value lv366

.±

0.01 A for the a:verage ri:ng bond

~ ~

dista.nca. With C-0

=

1.39

A , this corresponds to C-N

=

1.35

...! 0.03 A •

:s~atadiene and Cl_clopenta.dl_ep,e.- The photog:ra.p'hti of butadiene wh1.ch we1•0

f,l

Thi:rteen :rings could be seen, 1nth ·t:rio and

1

val ile s gi "fen

) ( >Ii• 4 0

l" :~ci A i's indi